The regulator is well fried!
You have to ask yourself when and how this happened, typically they make a pop, crack or hissing sound and smell bad when they fail :(. It is possible the part was faulty but it is also possible that it has been damaged by reversed power connection at the Vin header pin.
The regulator is pretty tough and tolerates most forms of abuse but not reversed polarity. The input jack has a diode in the power line that protects against reversed connection but that may be blown to a near short circuit if you have been probing the board and accidentally shorted out something (meter on Amps range?), it is certainly suspect as 3V is getting dropped somewhere according to your measurements.
So you will have to use the board on 5V USB (check the regulator stays cool when powered from USB only). Alternatively repair the board by replacing the regulator as already suggested by INTP and replace that suspect diode too.